Geodesic
A Dirichlet-to-Neumann map for the Allen–Cahn equation on manifolds with boundary
Jared Marx-Kuo1 orcid
1Rice University, Houston, USA
Interfaces and free boundaries, Tome 27 (2025) no. 4, pp. 575-618

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DOI

We study the asymptotic behavior of Dirichlet minimizers to the Allen–Cahn equation on manifolds with boundary, and we relate the Neumann data to the geometry of the boundary. We show that Dirichlet minimizers are asymptotically local in orders of ε and compute expansions of the solution to high order. A key tool is showing that the linearized Allen–Cahn operator is invertible at the heteroclinic solution, on functions with 0 boundary condition. We apply our results to separating hypersurfaces in closed Riemannian manifolds. This gives a projection theorem about Allen–Cahn solutions near minimal surfaces, as constructed by Pacard–Ritoré.
DOI : 10.4171/ifb/550
Classification : 53A10, 35G30
Mots-clés : Allen–Cahn, Dirichlet, Neumann

Jared Marx-Kuo  1

1 Rice University, Houston, USA 
Jared Marx-Kuo. A Dirichlet-to-Neumann map for the Allen–Cahn equation on manifolds with boundary. Interfaces and free boundaries, Tome 27 (2025) no. 4, pp. 575-618. doi: 10.4171/ifb/550
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     pages = {575--618},
     year = {2025},
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     doi = {10.4171/ifb/550},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/550/}
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